Mathematics Optional Applied Mathematics Pack

Cartesian and polar coordinates in three dimensions, second degree equations in three variables, reduction to canonical forms, straight lines, shortest distance between two skew lines; plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.

Formulation of differential equations; equations of first order and first degree, integrating factor; orthogonal trajectory; equations of first order but not of first degree, Clairaut's equation, singular solution; second and higher order linear equations with constant coefficients, complementary function, particular integral and general solution; second order linear equations with variable coefficients, Euler-Cauchy equation; complete solution when one solution is known, using the method of variation of parameters; Laplace and inverse Laplace transforms and their properties; application to initial value problems for second order linear equations with constant coefficients.

Family of surfaces in three dimensions and formulation of partial differential equations; solution of quasilinear partial differential equations of the first order, Cauchy's method of characteristics; linear partial differential equations of the second order with constant coefficients, canonical form; equation of a vibrating string, heat equation, Laplace equation and their solutions.

Analytic functions, Cauchy-Riemann equations, Cauchy's theorem, Cauchy's integral formula, power series representation of an analytic function, Taylor's series; singularities; Laurent's series; Cauchy's residue theorem; contour integration.

Numerical methods: bisection, regula-falsi and Newton-Raphson methods; Gaussian elimination, Gauss-Jordan and Gauss-Seidel methods; Newton's and Lagrange's interpolation; trapezoidal rule, Simpson's rule, Gaussian quadrature; Euler and Runge-Kutta methods. Computer programming: binary and hexadecimal systems, Boolean algebra, basic logic gates; representation of signed and unsigned integers and reals; algorithms and flowcharts for numerical problems.